Linux Memory Mapping

Purpose

The following examples demonstrates how to map a driver allocated buffer from kernel into user space. It has been tested with Linux kernel 2.2.18 and 2.4.0 on Intel (uni processor only) and Alpha platform (COMPAQ Personal Workstation 500au (uni processor), DS20 and ES40 (SMP).

Memory Mappings

The Linux kernel works with different memory mappings. One mapping, called kernel virtual mapping provides a direct 1 to 1 mapping of physical addresses to virtual addresses. Conversion between physical and virtual addresses can be made with phys_to_virt() and virt_to_phys() for the translation virtual to physical.
The translation from IO bus addresses into kernel virtual addresses uses also the kernel virutal mapping. An contiguous address area in the kernel segment is also contigous in physical memory.

Memory Allocation in the Kernel

kmalloc() returns a memory area in the kernel virtual mapping. Therefore, the area is physical contigous, and can be translated to a physical address with virt_to_phys() or to a IO bus address with virt_to_bus().
vmalloc() creates a new memory area, puts several physically non-contigous pages into the area and enters the new area into the memory map of the kernel. Such addresses cannot be converted into physical or IO bus addresses with the functions described above.

Translation Virtual to Kernel Virtual Address

Before translating into a physical address or into a IO bus address, a general kernel virtual address, e.g. returned by vmalloc(), must be converted to a kernel virtual address. This can be achieved e.g. with the following steps: Note: the function performs the translation for the *one* page, where address is in. Since vmalloc'd areas do not have to be physical contigous, the next page may have a complete different offset!
Note: the parsing of the page table as described above works when they do not change during the parsing. For memory areas allocated with vmalloc() this is the case. If you want to translate an address belong to a process which can get swapped out, you need to protect the code with the corresponding locks. See e.g. sys_mlock() how this can be done.

Mapping Kernel Virtual Addresses into User Space

2.2 and 2.4 Kernel

Mapping addresses which are in the kernel virtual mapping into user space is straight foreward: Example:
  vma->flags |= VM_LOCKED;
  if (remap_page_range(vma->vm_start,
                     virt_to_phys((void*)((unsigned long)kmalloc_area)),
                     size,
                     PAGE_SHARED))
{
     printk("remap page range failed\n");
     return -ENXIO;
}
Note: on Linux 2.4.x remap_page_range() needs to be called with the mm semaphore hold. The semaphore is grabed by the syscall, so within your mmap method you are safe. If you call remap_page_range in other contexts, you need to grab the semaphore first (e.g. down(&current->mm->mmap_sem)).

2.6 Kernel

On 2.6 things got even simpler. The remap_pfn_range function sets the correct flags in the vm_area. remap_pfn_range can be called for a set of physically contiguous pages. Do map the pages you therefore have to: Example:
	if (remap_pfn_range(vma,
			    vma->vm_start,
			    virt_to_phys((void *)kmalloc_area) >> PAGE_SHIFT,
			    size,
			    vma->vm_page_prot)) < 0) {
		printk("remap_pfn_range failed\n");
		return -EIO;
	}
The arguments of the remap_pfn_range function are:

Mapping non-kernel Virtual Addresses into User Space

Mapping addresses e.g. returned by vmalloc() into user space is a little bit more tricky, since each page has a different address translation.
A very elegant method to create such mappings is the usage of the nopage method of the virtual memory area functions. The methods are attached to the virtual memory area in the mmap method of the device. Each time than the user space process accesses a page that has not yet been translated, a page fault occurs and our nopage handler is called. The nopage handler has to increment the usage count of the page. On Linux 2.2.x it has to return the kernel virtual address of the page the application wants to access, on Linux 2.4.x it has to return the pointer to the page structure.
Example for a nopage handler:
Note: virt_to_kseg() is an implementation of the function described above to parse the page table.
/* page fault handler (for Linux 2.2.x) */
unsigned long mmap_drv_vmmap(struct vm_area_struct *vma, unsigned long address, int no_share)
{
        unsigned long offset;
        unsigned long virt_addr;

         /* determine the offset within the vmalloc'd area  */
        offset = address - vma->vm_start + vma->vm_offset;

        /* calculate the kseg virtual address */
        virt_addr = (unsigned long)virt_to_kseg(&vmalloc_area[offset/sizeof(int)]);

        /* increment the usage count of the page */
        atomic_inc(&mem_map[MAP_NR(virt_addr)].count);
        
        printk("mmap_drv: page fault for offset 0x%lx (kseg x%lx)\n",
               offset, virt_addr);

        /* return the kseg virtual address, *not* the physical address as stated
           in some wrong examples.
        */
        return(virt_addr);
}

2.6 Kernel

On 2.6 there is no need for a driver specific page fault handler since remap_pfn_range can be called for every page individually. To map a vmalloc'd area you simply have to loop over all pages and call remap_pfn_range:
        while (length > 0) {
                pfn = vmalloc_to_pfn(vmalloc_area_ptr);
                if ((ret = remap_pfn_range(vma, start, pfn, PAGE_SIZE,
                                           PAGE_SHARED)) < 0) {
                        return ret;
                }
                start += PAGE_SIZE;
                vmalloc_area_ptr += PAGE_SIZE;
                length -= PAGE_SIZE;
        }

Setting the Reserved Bit

Before a page can be exported into user space, the reserved bit must be set. This is done on Linux 2.2.x with e.g.:
mem_map_reserve(MAP_NR(virt_to_kseg((void *)virt_addr))) Note: mem_map_reserve() (and its counterpart mem_map_unreserve()) take the map number of the page as argument. The map number is calculated out of the kernel virtual address with the MAP_NR() macro.
On Linux 2.4.x mem_map_reserve() takes a pointer to a page structure as argument. The page structure pointer is derived from the kernel virtual address with virt_to_page().

Putting the Parts together

The example below shows a device driver, that allocates two memory area: one with vmalloc(), the other with kmalloc(). It implements both mapping methods described above to export the memory to user space.
Please read the explanations in the example program source code on how to run the test program.

Linux 2.6 Device Driver

With Linux 2.6 a new build process is used. This and the amount of changes made me split the example from the 2.2 and 2.4 code. Please download it from here.
A pure BSD licensed version of the code can be fetched from here.

Linux 2.2 and 2.4 Device Driver

The example has been tested with Linux 2.2.18 and 2.4.0, on Intel and Alpha platform. (File mmap_drv.c)
#include <linux/config.h>
#include <linux/version.h>
#include <linux/module.h>
#if defined(MODVERSIONS)
#include <linux/modversions.h>
#endif

#include <linux/kernel.h>
#include <linux/fs.h>
#include <linux/string.h>
#include <linux/module.h>
#include <linux/errno.h>

#include <linux/mm.h>
#include <linux/vmalloc.h>
#include <linux/mman.h>
#include <linux/wrapper.h>
#include <linux/slab.h>
#include <asm/io.h>

#define LEN (64*1024)

/* device open */
int mmapdrv_open(struct inode *inode, struct file *file);
/* device close */
int mmapdrv_release(struct inode *inode, struct file *file);
/* device mmap */
int mmapdrv_mmap(struct file *file, struct vm_area_struct *vma);

/* open handler for vm area */
void mmap_drv_vopen(struct vm_area_struct *vma);
/* close handler form vm area */
void mmap_drv_vclose(struct vm_area_struct *vma);
/* page fault handler for callback of vmalloc area */
#if LINUX_VERSION_CODE < KERNEL_VERSION(2,4,0)
unsigned long mmap_drv_vmmap(struct vm_area_struct *vma, unsigned long address, int write_access);
#else
struct page *mmap_drv_vmmap(struct vm_area_struct *vma, unsigned long address, int write_access);
#endif

/* the ordinary device operations */
static struct file_operations mmapdrv_fops =
{
#if LINUX_VERSION_CODE >= KERNEL_VERSION(2,4,0)
  owner:   THIS_MODULE,
#endif
  mmap:    mmapdrv_mmap,
  open:    mmapdrv_open,
  release: mmapdrv_release,
};

/* memory handler functions */
static struct vm_operations_struct mmap_drv_vm_ops = {
  open:    mmap_drv_vopen, /* mmap-open */
  close:  mmap_drv_vclose,/* mmap-close */
  nopage: mmap_drv_vmmap, /* no-page fault handler */
};

/* pointer to page aligned area */
static int *vmalloc_area = NULL;
/* pointer to unaligend area */
static int *vmalloc_ptr  = NULL;
/* pointer to page aligned area */
static int *kmalloc_area = NULL;
/* pointer to unaligned area */
static int *kmalloc_ptr = NULL;
/* major number of device */
static int major;

#if LINUX_VERSION_CODE < KERNEL_VERSION(2,4,0)
/* Converts a kernel virtual address into a kernel virtual
   address that is part of the direct mapping between
   virtual and physical address. If you e.g. allocated
   memory with vmalloc(), you get virtual addresses part
   of an own area. By converting such an address, 
   you receive a kernel virtual address that you can
   e.g. feed into virt_to_phys() or MAP_NR().
   Note: the function below works for one page. If you
   have a set of pages, in a vmalloc allocated area,
  each page may have a different virtual address in
   the direct mapping.
   Return 0 if no mapping found.
*/
volatile void *virt_to_kseg(volatile void *address)
{
        pgd_t *pgd; pmd_t *pmd; pte_t *ptep, pte;
        unsigned long ret=0UL;
	
        /* if we are below the max direct mappings, we use the
           direct conversion function
        */ 
        if (MAP_NR(address) < max_mapnr)
                return(address);

        /* else we really have to parse the page table to get the map nr */

	/* get the page global directory out of the kernel memory map. */
	pgd = pgd_offset_k((unsigned long)address);

	/* check whether we found an entry */
	if (!pgd_none(*pgd))
        {
	       /* get the page middle directory */
	       pmd = pmd_offset(pgd, (unsigned long)address);
	       /* check for a valid entry */
	       if (!pmd_none(*pmd))
               {
		    /* get a pointer to the page table entry */
	            ptep = pte_offset(pmd, (unsigned long)address);
		    /* get the page table entry itself */
	            pte = *ptep;
		    /* check for a valid page */
	            if (pte_present(pte))
                    {
		      /* get the kseg address of the page */
		      ret = (unsigned long)pte_page(pte);
		      /* add the offset within the page to the page address */
		      ret |= ((unsigned long)address & (PAGE_SIZE - 1));
		    }
	       }
	}
        return((volatile void *)ret);
}
#else
/* we parse the page tables in order to find the direct mapping of
   the page. This works only without holding any locks for pages we
   are sure that they do not move in memory.
   Annother example achieving the same can be found in the
   bttv-driver (drivers/media/video).
*/
volatile void *virt_to_kseg(volatile void *address)
{
        pgd_t *pgd; pmd_t *pmd; pte_t *ptep, pte;
	unsigned long va, ret = 0UL;
	
	va=VMALLOC_VMADDR((unsigned long)address);
	
	/* get the page directory. Use the kernel memory map. */
	pgd = pgd_offset_k(va);

	/* check whether we found an entry */
	if (!pgd_none(*pgd))
        {
	      /* get the page middle directory */
	      pmd = pmd_offset(pgd, va);
	      /* check whether we found an entry */
	      if (!pmd_none(*pmd))
              {
		  /* get a pointer to the page table entry */
	          ptep = pte_offset(pmd, va);
	          pte = *ptep;
		  /* check for a valid page */
	          if (pte_present(pte))
                  {
		        /* get the address the page is refering to */
		        ret = (unsigned long)page_address(pte_page(pte));
			/* add the offset within the page to the page address */
			ret |= (va & (PAGE_SIZE -1));
		  }
	      }
	}
	return((volatile void *)ret);
}
#endif

/* load the module */
int init_module(void)
{
        int i;
        unsigned long virt_addr;
        
        if ((major=register_chrdev(0, "mmapdrv", &mmapdrv_fops))<0) {
                printk("mmapdrv: unable to register character device\n");
                return (-EIO);
        }

        /* get a memory area with kmalloc and aligned it to a page. This area
           will be physically contigous */
        kmalloc_ptr=kmalloc(LEN+2*PAGE_SIZE, GFP_KERNEL);
        kmalloc_area=(int *)(((unsigned long)kmalloc_ptr + PAGE_SIZE -1) & PAGE_MASK);
#if LINUX_VERSION_CODE < KERNEL_VERSION(2,4,0)
       for(i=MAP_NR(kmalloc_area); i<=MAP_NR((void *)kmalloc_area+LEN);i++)
       {
                /* reserve all pages to make them remapable */
                mem_map_reserve(i);
       }
#else
       for (virt_addr=(unsigned long)kmalloc_area; virt_addr<(unsigned long)kmalloc_area+LEN;
	    virt_addr+=PAGE_SIZE)
       {
	        /* reserve all pages to make them remapable */
	        mem_map_reserve(virt_to_page(virt_addr));
       }
#endif

        /* get a memory area that is only virtual contigous. */
        vmalloc_ptr=vmalloc(LEN+2*PAGE_SIZE);
        vmalloc_area=(int *)(((unsigned long)vmalloc_ptr + PAGE_SIZE -1) & PAGE_MASK);
        for (virt_addr=(unsigned long)vmalloc_area;
	     virt_addr<(unsigned long)(&(vmalloc_area[LEN/sizeof(int)]));
             virt_addr+=PAGE_SIZE)
        {
#if LINUX_VERSION_CODE < KERNEL_VERSION(2,4,0)
                /* reserve all pages to make them remapable. */
                mem_map_reserve(MAP_NR(virt_to_kseg((void *)virt_addr)));
#else
		mem_map_reserve(virt_to_page(virt_to_kseg((void *)virt_addr)));
#endif
        }
        
        for (i=0; i<(LEN/sizeof(int)); i+=2)
        {
                /* initialise with some dummy values to compare later */
                vmalloc_area[i]=(0xaffe<<16) + i;
                vmalloc_area[i+1]=(0xbeef<<16) + i;
                kmalloc_area[i]=(0xdead<<16) +i;
                kmalloc_area[i+1]=(0xbeef<<16) + i;
        }

        /* and tell the world what we did */
        printk("vmalloc_area at 0x%p (phys 0x%lx)\n", vmalloc_area,
               virt_to_phys((void *)virt_to_kseg(vmalloc_area)));
        printk("kmalloc_area at 0x%p (phys 0x%lx)\n", kmalloc_area,
               virt_to_phys((void *)virt_to_kseg(kmalloc_area)));

        return(0);
}

/* remove the module */
void cleanup_module(void)
{
#if LINUX_VERSION_CODE < KERNEL_VERSION(2,4,0)
        int i;
#endif
        unsigned long virt_addr;

        /* unreserve all pages */
#if LINUX_VERSION_CODE < KERNEL_VERSION(2,4,0)
        for(i=MAP_NR(kmalloc_area); i<=MAP_NR((void *)kmalloc_area+LEN);i++)
        {
                mem_map_unreserve(i);
        }
#else
        for(virt_addr=(unsigned long)kmalloc_area; virt_addr<(unsigned long)kmalloc_area+LEN;
	    virt_addr+=PAGE_SIZE)
        {
                mem_map_unreserve(virt_to_page(virt_addr));
        }
#endif
        for (virt_addr=(unsigned long)vmalloc_area;
	     virt_addr<(unsigned long)(&(vmalloc_area[LEN/sizeof(int)]));
             virt_addr+=PAGE_SIZE)
        {
#if LINUX_VERSION_CODE < KERNEL_VERSION(2,4,0)
                mem_map_unreserve(MAP_NR(virt_to_kseg((void *)virt_addr)));
#else
		mem_map_unreserve(virt_to_page(virt_to_kseg((void *)virt_addr)));
#endif
        }

        /* and free the two areas */
        if (vmalloc_ptr)
                vfree(vmalloc_ptr);
        if (kmalloc_ptr)
                kfree(kmalloc_ptr);

        /* unregister the device */
        unregister_chrdev(major, "mmapdrv");
        return;
}

/* device open method */
int mmapdrv_open(struct inode *inode, struct file *file)
{
        MOD_INC_USE_COUNT;
        return(0);
}

/* device close method */
int mmapdrv_release(struct inode *inode, struct file *file)
{
        MOD_DEC_USE_COUNT;
        return(0);
}

/* device memory map method */
/* 2.4.x: this method is called from do_mmap_pgoff, from
   do_mmap, from the syscall. The caller of do_mmap grabs
   the mm semaphore. So we are protected from races here.
*/
int mmapdrv_mmap(struct file *file, struct vm_area_struct *vma)
{
#if LINUX_VERSION_CODE < KERNEL_VERSION(2,4,0)
        unsigned long offset = vma->vm_offset;
#else
	unsigned long offset = vma->vm_pgoff<<PAGE_SHIFT;
#endif
        unsigned long size = vma->vm_end - vma->vm_start;
        
        if (offset & ~PAGE_MASK)
        {
                printk("offset not aligned: %ld\n", offset);
                return -ENXIO;
        }
        
        if (size>LEN)
        {
                printk("size too big\n");
                return(-ENXIO);
        }
        
	/* we only support shared mappings. Copy on write mappings are
	   rejected here. A shared mapping that is writeable must have the
	   shared flag set.
	*/
	if ((vma->vm_flags & VM_WRITE) && !(vma->vm_flags & VM_SHARED))
	{
	     printk("writeable mappings must be shared, rejecting\n");
	     return(-EINVAL);
	}

	/* we do not want to have this area swapped out, lock it */
	vma->vm_flags |= VM_LOCKED;
        
        /* there are two different mapping options implemented here:
           for the virtual contiguous memory area, we install a page fault handler.
           The page fault handler calculates the right physical page on first
           access of the application to the page.
           (method 1 is used for vmalloc'd memory, offset 0..LEN)
           The second way works only for a physical contigous range of pages:
           we create a mapping between the physical pages and the virtual
           addresses of the application with remap_page_range.
           (method 2 is used for kmalloc'd memory, offset LEN..2*LEN)
        */
        if (offset == 0)
        {
                /* method 1: install a page handler */
                vma->vm_ops = &mmap_drv_vm_ops;
                /* call the open routine to increment the usage count */
                mmap_drv_vopen(vma);
        } else if (offset == LEN)
        {
                /* method 2: enter pages into mapping of application */
                if (remap_page_range(vma->vm_start,
                                     virt_to_phys((void*)((unsigned long)kmalloc_area)),
                                     size,
                                     PAGE_SHARED))
                {
                        printk("remap page range failed\n");
                        return -ENXIO;
                }
        } else
        {
                printk("offset out of range\n");
                return -ENXIO;
        }
        return(0);
}

/* open handler for vm area */
void mmap_drv_vopen(struct vm_area_struct *vma)
{
        /* needed to prevent the unloading of the module while
           somebody still has memory mapped */
        MOD_INC_USE_COUNT;
}

/* close handler form vm area */
void mmap_drv_vclose(struct vm_area_struct *vma)
{
        MOD_DEC_USE_COUNT;
}

/* page fault handler */
#if LINUX_VERSION_CODE < KERNEL_VERSION(2,4,0)
unsigned long mmap_drv_vmmap(struct vm_area_struct *vma, unsigned long address, int write_access)
#else
struct page *mmap_drv_vmmap(struct vm_area_struct *vma, unsigned long address, int write_access)
#endif
{
        unsigned long offset;
        unsigned long virt_addr;

         /* determine the offset within the vmalloc'd area  */
#if LINUX_VERSION_CODE < KERNEL_VERSION(2,4,0)
        offset = address - vma->vm_start + vma->vm_offset;
#else
        offset = address - vma->vm_start + (vma->vm_pgoff<<PAGE_SHIFT);
#endif

        /* calculate the kseg virtual address */
        virt_addr = (unsigned long)virt_to_kseg(&vmalloc_area[offset/sizeof(int)]);

	/* check whether we found a translation */
	if (virt_addr == 0UL)
	{
	       printk("page fault out of range\n");
#if LINUX_VERSION_CODE < KERNEL_VERSION(2,4,0)
	       return(virt_addr);
#else
	       return((struct page *)0UL);
#endif
	}

        /* increment the usage count of the page */
#if LINUX_VERSION_CODE < KERNEL_VERSION(2,4,0)
        atomic_inc(&mem_map[MAP_NR(virt_addr)].count);
#else
	atomic_inc(&(virt_to_page(virt_addr)->count));
#endif
        
        printk("mmap_drv: page fault for offset 0x%lx (kseg x%lx)\n",
               offset, virt_addr);

#if LINUX_VERSION_CODE < KERNEL_VERSION(2,4,0)
        /* return the kseg virtual address, *not* the physical address as stated
           in some wrong examples.
        */
        return(virt_addr);
#else
	/* return the page pointer */
	return(virt_to_page(virt_addr));
#endif
}

Test Application

(File mmap.c)
#include <stdio.h>
#include <unistd.h>
#include <sys/mman.h>
#include <sys/types.h>
#include <sys/stat.h>
#include <fcntl.h>

#define LEN (64*1024)

/* this is a test program that opens the mmap_drv.
   It reads out values of the kmalloc() and vmalloc()
   allocated areas and checks for correctness.
   You need a device special file to access the driver.
   The device special file is called 'node' and searched
   in the current directory.
   To create it
   - load the driver
     'insmod mmap_mod.o'
   - find the major number assigned to the driver
     'grep mmapdrv /proc/devices'
   - and create the special file (assuming major number 254)
     'mknod node c 254 0'
*/

int main(void)
{
  int fd;
  unsigned int *vadr;
  unsigned int *kadr;

  if ((fd=open("node", O_RDWR))<0)
    {
      perror("open");
      exit(-1);
    }
  vadr = mmap(0, LEN, PROT_READ, MAP_SHARED, fd, 0);
  
  if (vadr == MAP_FAILED)
  {
          perror("mmap");
          exit(-1);
  }
  if ((vadr[0]!=0xaffe0000) || (vadr[1]!=0xbeef0000)
      || (vadr[LEN/sizeof(int)-2]!=(0xaffe0000+LEN/sizeof(int)-2))
      || (vadr[LEN/sizeof(int)-1]!=(0xbeef0000+LEN/sizeof(int)-2)))
  {
       printf("0x%x 0x%x\n", vadr[0], vadr[1]);
       printf("0x%x 0x%x\n", vadr[LEN/sizeof(int)-2], vadr[LEN/sizeof(int)-1]);
  }

  kadr = mmap(0, LEN, PROT_READ, MAP_SHARED, fd, LEN);
  
  if (kadr == MAP_FAILED)
  {
          perror("mmap");
          exit(-1);
  }
  if ((kadr[0]!=0xdead0000) || (kadr[1]!=0xbeef0000)
      || (kadr[LEN/sizeof(int)-2]!=(0xdead0000+LEN/sizeof(int)-2))
      || (kadr[LEN/sizeof(int)-1]!=(0xbeef0000+LEN/sizeof(int)-2)))
  {
      printf("0x%x 0x%x\n", kadr[0], kadr[1]);
      printf("0x%x 0x%x\n", kadr[LEN/sizeof(int)-2], kadr[LEN/sizeof(int)-1]);
  }
  
  close(fd);
  return(0);
}

Makefile

When copy-pasting this makefile, remember the tabs on the start of the line!
Edit the first line of the makefile to adjust to your kernel source tree. You need to configure the kernel tree (e.g. make config) before to have a .config file created an have the symbolic links set up right.
# set to your kernel tree
KERNEL  = /usr/src/linux-2.4.0
#KERNEL  = /usr/src/linux-2.2.18

# get the Linux architecture. Needed to find proper include file for CFLAGS
ARCH=$(shell uname -m | sed -e s/i.86/i386/ -e s/sun4u/sparc64/ -e s/arm.*/arm/ -e s/sa110/arm/)
# set default flags to compile module
CFLAGS = -D__KERNEL__ -DMODULE -I$(KERNEL)/include
CFLAGS+= -Wall -Wstrict-prototypes -O2 -fomit-frame-pointer -fno-strict-aliasing

all:	mmap_mod.o mmap

# get configuration of kernel
include $(KERNEL)/.config
# modify CFLAGS with architecture specific flags
include $(KERNEL)/arch/${ARCH}/Makefile

# enable the module versions, if configured in kernel source tree
ifdef CONFIG_MODVERSIONS
CFLAGS+= -DMODVERSIONS -include $(KERNEL)/include/linux/modversions.h
endif
# enable SMP, if configured in kernel source tree
ifdef CONFIG_SMP
CFLAGS+= -D__SMP__
endif

# note: we are compiling the driver object file and then linking
# we link it into the module. With just one object file as in
# this example this is not needed. We can just load the object
# file produced by gcc 

# link the mmap driver module
mmap_mod.o:	mmap_drv.o
	ld -r -o mmap_mod.o mmap_drv.o

# compile the mmap driver
mmap_drv.o:	mmap_drv.c
	gcc $(CFLAGS) -c mmap_drv.c
# compile and link the test program
mmap:	mmap.c
	gcc -o mmap mmap.c

clean:
	rm -f *.o mmap

Comments, Corrections

Please send comments, corrections etc. to the address below.
frey@scs.ch